Well, wood bowl shapes will flex. How much depends on several things. One is thickness. Another is size/diameter. Another is speed/revs/rpm. Another is bevel pressure. Oh, I do mostly green once turned bowls, so my views are skewed to that perspective.
The thinner the bowl shape is the more likely it is to flex as you turn out the inside. It will elongate along grain likes, longer in the up/down grain lines, less in the side grain, and with really wet once turned pieces, losing water can contribute to this as well. A one inch thick bowl will not flex as much as a 1/4 inch thick bowl.
Size matters. On smaller pieces, say 10 inches or so and down, it isn't too difficult to make one final cut from rim to base if you have the proper gouge nose angle, usually in the 60 degree range. The bowl may move a little as you remove bulk, but that is minimal. On deeper and bigger bowls, some where in the 4 or so inch deep and 12 or so inch diameter, Things begins to change. You end up having to turn out the inside in stages. You do down an inch or two, do what ever clean up is necessary (shear scrape/NRS/another very light cut) and then go down another inch or two, gently picking up the cut from where you left off. There is generally a little line at the pick up point because the bowl has moved.
The higher the speeds are, the more likely you are go get distortion in the bowl shape as it spins. If you are turning end grain, this is almost non existent. With side grain, due to issues of end grain/side grain, the piece will not spin true. Diameter and rpm can make pretty big differences.
An old quote I love to use, and have no idea who said it first, "The bevel should rub the wood, but the wood shouldn't know it." On the outside of the bowl, you can push the bevel into the wood fairly hard, but since there is a lot of mass on the other side, the wood doesn't move, but you do get burnishing of the wood. There is always a tendency to push on or into the bevel rather than pushing into the cut. On the inside of the bowl, this can make a huge difference. You have no mass pushing back to you, so if you are pushing into the bevel rather than into the cut, you will get a lot of movement. In this case, wall thickness/thinness as well as grain orientation will cause the bowl to flex. Probably the most difficult thing to learn is to float the bevel as Stuart Batty says. The whole thing about turning in general is that it doesn't take brute strength to turn. If you are having to use that, then your tools are most likely dull.
Perhaps one final point, I think green wood flexes a lot more than dry wood. This also could be partly due to moisture loss during turning.
robo hippy